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Properties

Label 16.0.51399544780...7056.8
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 7^{8}$
Root discriminant $17.06$
Ramified primes $2, 3, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2\wr C_2$ (as 16T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![133, -630, 974, -338, -82, 888, -850, 378, -32, -96, 166, -118, 75, -40, 16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 16*x^14 - 40*x^13 + 75*x^12 - 118*x^11 + 166*x^10 - 96*x^9 - 32*x^8 + 378*x^7 - 850*x^6 + 888*x^5 - 82*x^4 - 338*x^3 + 974*x^2 - 630*x + 133)
 
gp: K = bnfinit(x^16 - 4*x^15 + 16*x^14 - 40*x^13 + 75*x^12 - 118*x^11 + 166*x^10 - 96*x^9 - 32*x^8 + 378*x^7 - 850*x^6 + 888*x^5 - 82*x^4 - 338*x^3 + 974*x^2 - 630*x + 133, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 16 x^{14} - 40 x^{13} + 75 x^{12} - 118 x^{11} + 166 x^{10} - 96 x^{9} - 32 x^{8} + 378 x^{7} - 850 x^{6} + 888 x^{5} - 82 x^{4} - 338 x^{3} + 974 x^{2} - 630 x + 133 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(51399544780206637056=2^{24}\cdot 3^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.06$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{9} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2}$, $\frac{1}{21} a^{12} + \frac{1}{21} a^{10} + \frac{10}{21} a^{9} - \frac{2}{7} a^{8} + \frac{1}{3} a^{6} + \frac{1}{7} a^{5} - \frac{2}{21} a^{4} - \frac{1}{21} a^{3} - \frac{2}{7} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{21} a^{13} + \frac{1}{21} a^{11} + \frac{1}{21} a^{10} + \frac{1}{7} a^{9} + \frac{4}{21} a^{7} - \frac{1}{7} a^{6} + \frac{10}{21} a^{5} - \frac{1}{21} a^{4} + \frac{2}{7} a^{3} - \frac{8}{21} a^{2} + \frac{1}{3} a$, $\frac{1}{1911} a^{14} - \frac{1}{273} a^{13} - \frac{3}{637} a^{12} + \frac{12}{637} a^{11} - \frac{74}{1911} a^{10} + \frac{905}{1911} a^{9} + \frac{673}{1911} a^{8} - \frac{61}{147} a^{7} - \frac{255}{637} a^{6} - \frac{62}{273} a^{5} - \frac{10}{637} a^{4} - \frac{898}{1911} a^{3} + \frac{87}{637} a^{2} - \frac{44}{273} a + \frac{38}{273}$, $\frac{1}{6012090480779019} a^{15} - \frac{5703188560}{95430007631413} a^{14} - \frac{56928377617316}{6012090480779019} a^{13} - \frac{30834002456966}{2004030160259673} a^{12} + \frac{17524518394416}{668010053419891} a^{11} + \frac{223491646739741}{6012090480779019} a^{10} + \frac{20344380994521}{51385388724607} a^{9} - \frac{730769050902425}{2004030160259673} a^{8} + \frac{53241779729647}{6012090480779019} a^{7} + \frac{319365537571468}{858870068682717} a^{6} + \frac{164776794564080}{668010053419891} a^{5} + \frac{895068805811177}{2004030160259673} a^{4} - \frac{277353937858219}{6012090480779019} a^{3} + \frac{52280787139676}{286290022894239} a^{2} - \frac{89691014804857}{858870068682717} a + \frac{17107774553060}{122695724097531}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{578677982}{63360528637} a^{15} + \frac{6760389338}{190081585911} a^{14} - \frac{27394244368}{190081585911} a^{13} + \frac{68298266545}{190081585911} a^{12} - \frac{131079577922}{190081585911} a^{11} + \frac{215042047762}{190081585911} a^{10} - \frac{324716234686}{190081585911} a^{9} + \frac{246559333526}{190081585911} a^{8} - \frac{16641398668}{27154512273} a^{7} - \frac{396570950432}{190081585911} a^{6} + \frac{373654627306}{63360528637} a^{5} - \frac{59008987591}{9051504091} a^{4} + \frac{39203334656}{63360528637} a^{3} + \frac{5447445352}{190081585911} a^{2} - \frac{137017984202}{27154512273} a + \frac{65469261755}{27154512273} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4124.94709562 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2$ (as 16T39):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-21}) \), 4.0.3024.2, 4.0.1008.2 x2, 4.2.9408.1 x2, 4.0.189.1, 4.0.21168.1, \(\Q(\sqrt{-3}, \sqrt{7})\), 4.0.432.1, 8.0.7169347584.5, 8.0.7169347584.4, 8.0.448084224.2, 8.0.448084224.7, 8.0.796594176.7, 8.0.9144576.1, 8.0.146313216.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$